The generic symplectic \(C^1\)-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point. (English) Zbl 1030.37037

Arnaud, Marie-Claude

Ergodic Theory Dyn. Syst. 22, No. 6, 1621-1639 (2002).

The title of this paper reflects well the main result. More precisely, a refinement of a result of S. E. Newhouse [Am. J. Math. 99, 1061-1087 (1977; Zbl 0379.58011)] is given in a four-dimensional case.

Reviewer: Jan Andres (Olomouc)

MSC:

37J10	Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37C05	Dynamical systems involving smooth mappings and diffeomorphisms
37C20	Generic properties, structural stability of dynamical systems
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D30	Partially hyperbolic systems and dominated splittings
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics

Keywords:

symplectic diffeomorphisms; four-dimensional manifolds; generic results